Body Force Propeller Method

The body force propeller method is used to model the effects of a propeller such as thrust and torque and thereby creating propulsion without actually resolving the geometry of the propeller. The method employs a uniform volume force $f_{b}$ distribution over the cylindrical virtual disk. The volume force varies in radial direction.

The radial distribution of the force components follows the Goldstein optimum and is given by:

Figure 1. EQUATION_DISPLAY

f_{b x} = A_{x} r^{*} \sqrt{1 - r^{*}}

(4976)

Figure 2. EQUATION_DISPLAY

f_{b θ} = A_{θ} \cdot \frac{r^{*} \sqrt{1 - r^{*}}}{r^{*} (1 - {r′}_{h}) + {r′}_{h}}

(4977)

Figure 3. EQUATION_DISPLAY

r^{*} = \frac{r′ - {r′}_{h}}{1 - {r′}_{h}}

(4978)

{r′}_{h} = \frac{R_{H}}{R_{P}} and r′ = \frac{r}{R_{P}}

where $f_{b x}$ is the body force component in axial direction, $f_{b θ}$ is the body force component in tangential direction, r is the radial coordinate, $R_{H}$ is the hub radius and $R_{P}$ is the propeller tip radius.

The constants $A_{x}$ and $A_{θ}$ are computed as:

Figure 4. EQUATION_DISPLAY

A_{x} = \frac{105}{8} \cdot \frac{T}{π Δ (3 R_{H} + 4 R_{P}) (R_{P} - R_{H})}

(4979)

Figure 5. EQUATION_DISPLAY

A_{θ} = \frac{105}{8} \cdot \frac{Q}{π Δ R_{P} (3 R_{H} + 4 R_{P}) (R_{P} - R_{H})}

(4980)

where T is the thrust, Q is the torque, and $Δ$ is the virtual disk thickness.

The computation of the body force components requires several user inputs. A propeller performance curve needs to be specified, which gives the dimensionless thrust coefficient $K_{T}$ , the torque coefficient $K_{Q}$ , and the propeller efficiency $η_{0}$ as functions of the advance ratio J.

Figure 6. EQUATION_DISPLAY

K_{T} = \frac{T}{ρ n^{2} {D_{p}}^{4}}

(4981)

Figure 7. EQUATION_DISPLAY

K_{Q} = \frac{Q}{ρ n^{2} {D_{p}}^{5}}

(4982)

Figure 8. EQUATION_DISPLAY

J = \frac{V_{A}}{n D_{P}}

(4983)

where $V_{A}$ is the speed of advance of the propeller, $n$ the rotation rate with the unit revolution per second $[r p s]$ , and $D_{P}$ the propeller diameter.

Further inputs are the position of the propeller within the computational domain, the direction of the propeller rotational axis, and the direction of rotation.

The simulation is performed for a certain operating point. The operating point can be specified by either of these quantities:

Rotation rate n
Thrust T
Torque Q

If the operation point is given by rotation rate n, the procedure for obtaining the body force component distribution over the virtual disk is the following:

The advance ratio J is calculated as:
Figure 9. EQUATION_DISPLAY

$J = \frac{V_{Inflow Plane}}{n D_{P}}$
(4984)
The thrust coefficient $K_{T}$ and the torque coefficient $K_{Q}$ are interpolated from the specified propeller performance curve:
$K_{T}, K_{Q} = f (J)$
With $K_{T}$ and $K_{Q}$ available, thrust T and torque Q are calculated for the propeller:
Figure 10. EQUATION_DISPLAY

$T = \frac{K_{T} ρ {V^{2}}_{Inflow Plane} {D^{2}}_{P}}{J^{2}}$
(4985)
Figure 11. EQUATION_DISPLAY

$Q = \frac{K_{Q} ρ {V^{2}}_{Inflow Plane} {D^{3}}_{P}}{J^{2}}$
(4986)
With T and Q available, the axial and the tangential body force components $f_{b x}$ and $f_{b θ}$ are calculated according to Eqn. (4976) to Eqn. (4980).

If the operation point is given by thrust T or by torque Q, the first step of above procedure is replaced by:

With a given T or Q, the advance ratio is calculated by solving the following equation numerically:
Figure 12. EQUATION_DISPLAY

$f (J) = K_{T} - {K′}_{T} or f (J) = K_{Q} - {K′}_{Q}$
(4987)

where $K_{T}$ and $K_{Q}$ are evaluated from the propeller performance curve.

${K′}_{T}$ and ${K′}_{Q}$ are evaluated as:
Figure 13. EQUATION_DISPLAY

${K′}_{T} = \frac{J^{2} T_{Operating Point}}{ρ_{Inflow Plane} {V^{2}}_{Inflow Plane} {D_{P}}^{2}}$
(4988)
Figure 14. EQUATION_DISPLAY

${K′}_{Q} = \frac{J^{2} Q_{Operating Point}}{ρ_{Inflow Plane} {V^{2}}_{Inflow Plane} {D_{P}}^{3}}$
(4989)

With the advance ratio J available, the calculation of the axial and tangential body force components follows the same procedure as for the rotation rate operating point continuing with the second step.